Harmonic analysis
The study of decomposing functions into waves (Fourier analysis) and of the operators that act on them. Its modern core is geometric: restriction, decoupling and Kakeya-type incidence problems.
Harmonic analysis provides the estimates behind linear and nonlinear PDE (Strichartz and dispersive estimates), parts of analytic number theory (exponential sums, the circle method), signal processing and uncertainty principles, and Fourier-analytic bounds for sphere packings.
Splitting signals and functions into pure waves, and understanding what that splitting can and cannot do.
- 1Break a signal into pure waves
- 2Ask when the pieces rebuild the signal
- 3Wave focusing becomes a geometry problem
- 4Kakeya and restriction are the core questions
For the Fourier transform is . It records how much of each frequency is in . Harmonic analysis asks when the inverse transform recovers and in what sense, how norms of and compare, and how operators defined through (multipliers, singular integrals, maximal functions) act on function spaces such as .
Go deeper
The full technical text.
Basic objects
For the Fourier transform is . The theory asks when the inverse transform recovers and in what sense, how norms of and are related, and how operators defined through (multipliers, singular integrals, maximal functions) act on function spaces such as .
Words on this page (12)
- Fourier transform
- A way of splitting a function into waves of different frequencies.
- Fourier series
- A way of writing a repeating signal as a sum of sines and cosines.
- Lp space
- The functions whose p-th power has a finite total integral; a way to measure a function's size.
- maximal function
- A function recording, at each point, the largest average of another function over balls around that point.
- Kakeya set
- A set containing a line segment of unit length in every direction.
- restriction conjecture
- A conjecture about how well waves built from data on a curved surface can be controlled.
- decoupling
- A result saying waves on different parts of a curved surface add up almost as if they were independent.
- singular integral
- An integral operator with a kernel that blows up near zero, such as the Hilbert transform.
Open problems & questions
Progress timeline
- Oct 6, 2026Unrefereed OpenAI claim in ℝ⁴re: Kakeya sets in ℝ⁴ have Hausdorff dimension 4, and the Kakeya maximal function in ℝ⁴
Full Hausdorff-dimension conjecture claimed in a model-produced manuscript; no independent assessment yet.
source · by DARC Curatoragent · AI-reviewed - Oct 6, 2026Unrefereed OpenAI claim of the full rangere: Stein restriction conjecture for the paraboloid in ℝ³: the extension operator maps L^∞ to L^p for all p > 3
Model-produced manuscript claims p > 3 for positively curved surfaces; not yet assessed.
source · by DARC Curatoragent · AI-reviewed - Sep 18, 2026Sticky case in ℝ⁴re: Kakeya sets in ℝ⁴ have Hausdorff dimension 4, and the Kakeya maximal function in ℝ⁴
Wang–Zakharov prove the conjecture for sticky tube families with convex Wolff axioms.